Close Binary Interactions of Intermediate-Mass Black Holes Possible Ultra-Luminous X-ray So

a r X i v :a s t r o -p h /0508597v 1 28 A u g 2005

S UBMITTED TO T HE A STROPHYSICAL J OURNAL ,A UGUST 27,2005

Preprint typeset using L A T E X style emulateapj v.6/22/04

CLOSE BINARY INTERACTIONS OF INTERMEDIATE-MASS BLACK HOLES:POSSIBLE ULTRA-LUMINOUS

X-RAY SOURCES?

L.B LECHA 1,N.I VANOVA 1,V.K ALOGERA 1,K.B ELCZYNSKI 2,J.F REGEAU 1,AND F.R ASIO 1

Submitted to The Astrophysical Journal,August 27,2005

ABSTRACT

While many observed ultra-luminous X-ray sources (ULXs,L X ≥1039erg s ?1)could be extragalactic X-ray binaries (XRBs)emitting close to the Eddington limit,the highest-luminosity ULXs (L X >3×1039erg s ?1)exceed the isotropic Eddington luminosity for even high-stellar-mass accreting black hole XRBs.It has been suggested that these highest-luminosity ULXs may contain accreting intermediate-mass black hole (IMBH)binaries.We consider this hypothesis for dense,young (～108yr)stellar clusters where we assume that a 50-500M ⊙central IMBH has formed through runaway growth of a massive https://www.360docs.net/doc/e16249956.html,ing numerical simulations of the dynamics and evolution of the central black hole’s captured companions,we obtain estimates of the incidence of mass transfer phases and possible ULX activity throughout the IMBH’s evolutionary history.We ?nd that,although it is common for the central black hole to acquire binary companions,there is a very low probability that these interacting binaries will become observable ULX sources.

Subject headings:Stars:binaries:close —X-rays:binaries —Black Holes —Star Clusters

1.INTRODUCTION

Extragalactic observations with the Einstein space X-ray telescope revealed a new category of sources,currently known as ultra-luminous X-ray sources (ULXs,Fabbiano 1989).In 1990,surveys conducted with the ROSAT satel-lite were able to resolve many of these sources as dis-tinct objects not associated with emission from galactic nu-clei.Up to half of the galaxies surveyed were found to contain off-nuclear ULXs (Colbert &Mushotzky 1999;Roberts &Warwick 2002;Lira,Lawrence,&Johnson 2000;Colbert &Ptak 2002;Colbert &Miller 2004).In analogy with X-ray sources observed in our own galaxy,many ULXs could be extragalactic X-ray binaries (XRBs)with stellar-mass black hole (BH)accretors (e.g.,Rappaport,Podsiad-lowski,&Pfahl 2005).However,in many cases their lu-minosities exceed the Eddington limit for isotropic emission from an accreting stellar-mass BH.The lowest luminosity that commonly de?nes a ULX,L X ≥1039erg s ?1,corresponds to an Eddington black hole mass 8M ⊙–still within the BH mass range allowed by stellar evolution models.Based on the X-ray survey conducted by Colbert &Ptak (2002),ULXs with L X ≥1039erg s ?1are estimated to exist in ～12%of all galaxies (Ptak &Colbert 2004).Of the 87X-ray (2-10keV)sources observed,45had L X >3×1039erg s ?1.This corre-sponds to the Eddington luminosity for the highest-mass black holes that can form through single-star collapse,～25M ⊙for a metallicity of Z =0.001(Belczynski,Sadowski,&Rasio 2004).We see that as much as half of this observed ULX pop-ulation requires an alternative to the model of isotropic emis-sion from a stellar-mass XRB.

Observations increasingly support the idea that some ULX sources may harbor intermediate-mass black holes (IMBHs,Colbert &Mushotzky 1999).Until recently,no evi-dence existed for BHs in the wide mass range between

1

Northwestern University,Dept of Physics &Astronomy,Evanston,IL 60208

l-blecha@https://www.360docs.net/doc/e16249956.html,;nata,vicky,fregeau,rasio@https://www.360docs.net/doc/e16249956.html, 2Tombaugh Fellow,Department of Astronomy,New Mexico State Uni-versity,Las Cruces,NM 88003kbelczyn@https://www.360docs.net/doc/e16249956.html,

stellar-mass and supermassive BHs.Many spectral and timing observations conducted with Chandra,ASCA,and XMM-Newton have con?rmed that ULXs are not associ-ated with supernovae and are consistent with accreting bi-naries (Fabbiano &White 2003).Additionally,ULX posi-tions are strongly correlated with young,star-forming re-gions.Examples include the nine ULXs found in the An-tennae (Fabbiano,Zezas,&Murray 2001)and the highest-luminosity ULX yet observed,in M82(peak L X ～9×1040erg s ?1,corresponding to an IMBH mass ?700M ⊙if L X ?L Edd X ;Kaaret et al.2001;Matsumoto et al.2001).The young,dense clusters where ULXs are often observed are also likely sites for IMBH formation (Freitag,Gürkan,&Rasio 2005;Freitag,Rasio,&Baumgardt 2005;Gürkan,Freitag,&Rasio 2004).

Accretion disk spectra provide a further test for the pres-ence of IMBHs:higher-mass BHs are expected to have cooler accretion disks modeled as multi-color disks (Mitsuda et al.1984).Some evidence for low-temperature disks was recently found using high-resolution XMM spectra of numerous ULXs,including those in NGC 1313,NGC 4559,Ho IX,and the Antennae (Miller et al.2003,2004a,b).As Colbert &Miller (2004)note,the sources observed to have cool disks are also the ones with luminosities too high to be explained by XRBs,even if they are beamed XRBs.Nevertheless,we note that X-ray spectral interpretation is subject to a number of model assumptions and may not be unique.

Anisotropic,or beamed,disk emission is another possible explanation for apparently super-Eddington X-ray luminosi-ties.The degree to which emission is beamed can be de?ned by the beaming fraction b =?/4π,where ?is the solid an-gle of emission.ULXs containing beamed stellar-mass XRBs could be a short phase of rapid mass transfer in the lifetime of ordinary XRBs,such as thermal-timescale mass transfer (King et al.2001).As demonstrated by King et al.(2001),by introducing mild beaming,the XRB falls within the stellar-mass BH range,assuming L X <1040erg s ?1.However,about 5-10%of the total ULX population is observed to have L X ≥1040erg s ?1(Colbert &Miller 2004;Ptak &Colbert 2004;Colbert &Ptak 2002).At these luminosities,very mas-sive BHs (～25M ⊙)could fall below the Eddington limit with

2Blecha et al.

b 0.25.To have the required mass ratio q =M 2/M 1>1,where M 1is the BH mass,these BHs would require a very massive companion.More moderate-mass BHs would require more severe beaming fractions that are not easily justi?ed.Furthermore,in at least two instances observations seem to indicate isotropi

c X-ray emission.These include an isotropic nebula observe

d arond th

e ULX in Ho II (Pakull &Mironi 2001)and the discovery o

f quasi-periodic oscillations (QPOs)from the brightest ULX in M82(Strohmayer &Mushotzky 2003).

Transient behavior of a ULX source is an important test for the presence of an IMBH (Kalogera et al.2004).If the ULX is a beamed stellar-mass XRB in thermal-timescale mass trans-fer,it is expected to produce persistent emission at or above the Eddington luminosity (King et al.2001).A persistent

IMBH ULX similarly would require a sustained ˙M compa-rable to the Eddington mass transfer rate:

˙M

Edd ?2.3×10?9

M 1M ⊙

?0.4 R 5×10?4

?0.5

g s ?1

(2)

for donor mass M 2and disk radius R.C is assumed to be

5×10?4for most donors.It can also be shown that the crit-ical ˙M

for transience corresponds to a minimum black hole mass that depends on companion mass and orbital period (King et al.1996).As Kalogera et al.(2004)have shown,this minimum BH mass is high for main sequence (MS)donor stars (103M ⊙on average for a 10M ⊙donor).Red giant (RG)donors can more easily form transient systems with IMBHs.One should note that,due to the minimum mass limit,a stellar-mass BH binary is unlikely to be transient,so any tran-sient ULX observed is a strong IMBH candidate.

In our simulations,we consider IMBHs that have formed through “runaway collisions”of MS stars.This is a channel for IMBH formation in which a series of rapid mergers occur as the cluster begins core col-lapse,causing growth of a central massive object that can collapse to a black hole (Portegies Zwart et al.1999;Portegies Zwart &McMillan 2002;Gürkan,Freitag,&Rasio 2004;Freitag,Rasio,&Baumgardt 2005;Freitag,Gürkan,&Rasio 2005).It has been sug-gested that an IMBH formed though runaway growth actually prevents core collapse and causes the core to reexpand (Baumgardt,Makino,&Ebisuzaki 2004).In

the dense clusters we are considering,the timescale for IMBH formation through runaway mergers is esti-mated to be 3Myr (Portegies Zwart &McMillan 2002;

Gürkan,Freitag,&Rasio 2004).Because the core collapse time,t cc ,is expected to obey the relation t cc ～0.15t rc (0),where t rc (0)is the initial core relaxation time,this constrains t rc to be 30Myr (Gürkan,Freitag,&Rasio 2004).

Given that a plausible IMBH formation method exists,the question most pertinent to understanding ULXs is whether an IMBH,once formed,will gain close stellar companions that can sustain mass transfer at ULX X-ray luminosities.No prior studies have attempted to investigate numerically with both dynamics and binary evolution whether binary formation and mass transfer can occur with IMBHs in a dense cluster core (for an analytical exploration of tidal capture and its conse-quences,see Hopman,Portegies Zwart,&Alexander 2004).Here,we take the ?rst step in studying IMBH binary inter-actions as a ULX possibility,using detailed numerical simu-lations that combine dynamical interactions with full binary stellar evolution.In §2we outline the details of our simu-lations.The results for our standard cluster model (details below)are presented in §3.The variations of these results for other cluster models are described in §4.In §5,we discuss our results,estimating a lower limit on the incidence of IMBH mass transfer,and we outline our goals for future simulations.

2.CLUSTER SIMULATIONS WITH CENTRAL IMBH

The goal of our simulations is to examine the ability of IMBHs in young,dense stellar clusters to form and main-tain close binary systems with companion stars.We em-ploy a numerical code developed by Ivanova et al.(2005,where a detailed description of the method can be found).This adopted hybrid code incorporates:(i)a stellar popula-tion synthesis code to evolve single stars and binary systems (StarTrack ;Belczynski,Kalogera,&Bulik 2002;Belczynski et al.2005to be submitted);(ii)a semi-analytical prescrip-tion for 2-,3-,and 4-body stellar collisions,disruptions,ex-changes,and tidal captures based on previously derived cross sections;and (iii)a numerical toolkit for direct N-body in-tegration of 3-and 4-body systems (Fewbody ;Fregeau et al.2004).This combination provides us with a unique numeri-cal tool that lends itself to the study of binary interactions in cluster cores.At the young cluster ages we consider,large-scale cluster evolution has little effect,so certain simpli?ca-tions can be made.We employ a “?xed background”model,which holds the core stellar number density n c and velocity dispersion σconstant (e.g.,Hut,McMillan,&Romani 1992;Di Stefano &Rappaport 1994;Sigurdsson &Phinney 1995;Portegies Zwart et al.1997a;Portegies Zwart et al.1997b;Rasio et al.2000).The cluster density pro?le is assigned by dividing the cluster into two regions:a dense core and an outer halo.This model is qualitatively based on mass segrega-tion.Note that the core is expected to expand on a very short timescale immediately after the IMBH is formed,smoothing out the cusp and supporting the assumption of a constant den-sity core (Baumgardt,Makino,&Ebisuzaki 2004).

In our models,an IMBH is added to the core at 1Myr,in accordance with estimated IMBH formation times of 1-3Myr in clusters (Portegies Zwart &McMillan 2002).To form an IMBH through runaway collisions,the initial core relaxation time,t rc ,must be 25?30Myr (Gürkan,Freitag,&Rasio 2004).This time is given by:

t rc =

σ33D

ULX Sources from Accreting Binaries With IMBH?3

TABLE1

C LUSTER M ODELS

Model IMF n c[pc?3]

A Kroupa1.33×105

B Flattened1.33×105

C Kroupa1.33×104

N OTE.—The IMF in models B

is a broken power-law mass func-

tionξ(M)～Mα,withα=-1.25

for0.8

for M/M⊙>5.Models A&B

were tested with and without3BBF.

Model A is also tested with a range

of IMBH masses from50to500

M⊙.

ber of stars, m is the average stellar mass,andγ,the co-

ef?cient in the Coulomb logarithmΛ=γN,is assumed to

be0.01(cf.Gürkan,Freitag,&Rasio2004).For our stan-

dard model(parameters are described below),t rc?29Myr,

in good agreement with the constraint for runaway collisions.

We evolve the clusters to100Myr,as we consider only

young clusters like those with which ULXs are most often

associated.We can then neglect long-term contributions of

the outer halo to cluster evolution and synthesize the entire

stellar population of N=2.7×104in the core.Mass segrega-

tion within the core is incorporated through our choice of the

initial mass function(IMF)used for the stellar population in

the core.

An initial binary fraction of100%is assumed,which pro-

vides the maximum possible concentration of binaries to in-

teract with and become companions of the IMBH.Observa-

tions indicate that young clusters are expected to have very

high binary fractions(Levine,Lada,&Elston1999;Apai

2004;

Delgado-Donate et al.2004).This assumption is also consistent with the suggestion that cluster binary popula-tions are depleted over time by interactions and stellar evolu-tion(Mikkola1983;Hills1984;Bacon,Sigurdsson,&Davies 1996;Fregeau et al.2004;Ivanova et al.2005).Initial condi-tions for binary orbital parameters are assigned as outlined in §3.1of Ivanova et al.(2005),with slight modi?cations:

?The binary mass ratio q=m2/m1is assigned a uniform

distribution0

primary and secondary,respectively.

?A uniform logarithmic distribution is used for the bi-

nary period,P,ranging from0.1?107d.

?The thermal distribution is used for the binary eccen-

tricities e with probability densityρ(e)=2e.?Binary systems are rejected from the initial distribution

if one of the binary components enters Roche lobe over-

?ow(RLOF)at pericenter.

?We use the IMF of Kroupa(2002)for primary stars be-

tween0.8and100M⊙in most simulations.?Secondary stars are restricted in a mass range between

0.05and100M⊙.

Our standard cluster model(model A,see Table1)has a1-D stellar velocity dispersionσ=10km s?1and a core stellar number density n c=1.33×105pc?3.The latter corresponds to a luminosity density≥107L⊙pc?3for a cluster age≤108 yr.The total cluster mass is?5×104M⊙.We use100M⊙as our standard IMBH mass,but we explore a range of IMBH masses from50?500M⊙(see§4.1).Note that each type of simulation must be repeated many times to obtain good statis-tics.

We also consider two variations of our standard cluster model(see Table1):with models B we explore the effect of a?atter IMF,and with models C we examine results for less dense clusters.Detailed simulations of clusters that form an IMBH just before core collapse have shown that rapid mass segregation preceding IMBH formation creates a mass func-tion?atter than Kroupa’s for stars>5?10M⊙(Gürkan,Fre-itag,&Rasio2004;Freitag,Gürkan,&Rasio2005;Freitag 2005,personal communication).These results have prompted us to consider the effects of mass segregation on the forma-tion of IMBH binaries.We have also examined whether any Brownian-like motion of the IMBH would cause it to inter-act with stellar populations that have different properties than those typical of the compact cluster cores.Assuming energy equipartition exists between the IMBH and the cluster,the IMBH in a constant-density core will experience radial oscil-lations of amplitude(Bahcall&Wolf1976)

R?

R core

3 M IMBH

M2 1/3.(5) We recognize that this arti?cial treatment can affect the evo-lution of IMBH binaries,for example,by preventing eccen-tricity boosts via the Kozai mechanism(Kozai1962).For this reason we have undertaken a careful analysis of triple break-ing and its effect on our results(see§3.5).It should also be noted that in some cases with a RG companion,release of the

4Blecha et al.

F IG.1.—Probability densities of IMBH binary properties throughout the simulations:companion mass,orbital separation and eccentricity.The ther-

mal distribution,ρ(e)=2e,is shown on the eccentricity plot.Results from

167simulations of model A with a100M⊙IMBH are shown.

outer companion induces a common-envelope phase in the re-

sulting binary.

3.RESULTS FOR STANDARD MODEL

In what follows we present the results of our standard,

100M⊙IMBH cluster simulations(model A in Table1)in

the context of the formation of IMBH binaries that experi-

ence mass transfer and may become observable as ULXs.In

§3.1we discuss the characteristics of IMBH companions,in-

cluding their orbital parameters,evolutionary types,and mass transfer phases.In§3.2we present an example of a single simulation and more details on the history of the IMBH.In

§3.3we examine the X-ray luminosities produced by mass transfer and determine the time fraction over which the IMBH

could be observable as a ULX.In§3.4we comment on the statistical interpretation of our results.Finally,in§3.5we an-

alyze triple stellar systems and their possible effect on mass-transferring binaries.

3.1.Statistical Results on IMBH Companions

The distributions of three orbital parameters:companion mass,orbital separation,and eccentricity,are shown in Fig.1

for all100M⊙IMBH simulations for cluster model A.The prevalence of low-mass companions(～1M⊙)in the core is consistent with the model IMF that is dominated by such low-

mass stars.Orbital separations of the dynamically-formed

IMBH binaries range widely from～1?107R⊙.The lower

limit corresponds to the Roche lobe radius of the IMBH com-panion.On the other end,the soft boundary for binaries is

de?ned as the“escape radius”

GM IMBH

a soft=

ULX Sources from Accreting Binaries With IMBH?

5

panion for only ?3%of the total evolution time (Table 2).Fig.2shows companion masses and orbital characteristics for different evolutionary types.The IMBH companions are overwhelmingly MS stars due to the much longer lifetime of this phase and their dominance by number.About 90%of unique 100M ⊙companions are MS stars for at least part of their binary lifetime;the IMBH has an average of ～6MS companions per simulation and only one post-MS compan-ion for every two simulations [“post-MS”will henceforth re-fer to all post-main-sequence evolutionary types capable of driving mass transfer -i.e.,neutron stars (NSs)and BHs are excluded].In contrast,post-MS companions are more likely candidates for RLOF than are MS companions.The percent-age of post-MS companions that undergo mass transfer in our simulations is much higher than the percentage of mass-transferring MS companions (see Table 2).As noted above,post-MS companions are also more likely to enter transient mass transfer phases,which are more conducive to ULX for-mation.However,because the MS lifetime is about 10times longer,mass transfer phases of MS companions are typically of longer duration,and the numbers of MS and post-MS mass-transferring companions per simulation are comparable.(Ta-ble 2).

3.2.Single Run Example

The binary companion history of a single 100M ⊙IMBH simulation,using model A,is shown in Figs.3&4.This sim-ulation is atypical in that it has multiple mass transfer events,but the behavior of these events is characteristic of mass trans-fer seen in other simulations.Fig.3shows the IMBH evolu-tion over time.The black hole has a companion for ～60%of its total evolution time,which is about the average we calcu-late for all 100M ⊙simulations (Table 2).Note the often high eccentricities of non-mass-transferring companions,which is also typical of our simulations (Fig.1).In the lower window of Fig.3,one can see the hardening of several companion or-bits before they are disrupted by another interaction.The long mass transfer phase at the end of the simulation characterizes stable,MS mass transfer.The ?rst companion that undergoes RLOF (at about 107yr)is the most interesting.Mass trans-fer begins on the MS and continues through the HG until the donor becomes a He HG star.After this,the star explodes and leaves a NS orbiting the IMBH.The companion’s circu-larized orbit is kicked by the supernova to an eccentricity of about 0.6.As Fig.3shows,the NS is later forced into an even higher-e orbit by an encounter with an outer compan-ion shown in Fig.4.This causes an eventual merger with the IMBH.The outer-companion swapping that occurs with the NS-IMBH binary is frequently-observed behavior for a close,stable binary;more details on the behavior of triple systems are presented in §3.5.

3.3.X-ray Luminosities

We examine the range of X-ray luminosities produced by the various mass-transfer phases that occur in our standard model simulations.The binary evolution component of the

simulations allows us to calculate the mass transfer rate ˙M

driven by RLOF donors to the IMBH.We compare this rate

to the critical rate ˙M

crit for transient behavior (Eqn.2)to de-termine whether the IMBH XRB is persistent or transient.If persistent,the transfer rate from the donor is converted to an expected X-ray luminosity;if transient,we assume that the XRB emits at its Eddington luminosity during disk outbursts

F I

G . 3.—Evolutionary history of a single 100M ⊙IMB

H simulation.Top window:IMBH companion mass M 2;middle window:eccentricity;bot-tom window:orbital separation.Circles denote timesteps with mass transfer;each non-mass-transferring companion appears at a different mass value.It is evident that the IMBH changes muliple binary companions during a single cluster simulation.Gaps in the top and bottom plots indicate times with no IMBH companion.

F I

G .4.—IMB

H companion mass (M 2)vs.orbital separation (a)for the same simulation shown in Fig.1.Dots denote timesteps with an IMBH com-panion;open circles denote mass-transferring companions.Triangles mark inner-binary orbital parameters of triple systems and solid lines connect to the outer orbital parameters.The maximum companion orbit radii for RLOF are shown at:the end of MS (line 1),the start of the RG branch (line 2),the maximum possible stellar radius (line 3),and the maximum stellar radius for clusters younger than 108yr (line 4).Line 5marks the soft/hard binary boundary.

6Blecha et al.

TABLE2

IMBH BINARY COMPANIONS

M IMBH=100M⊙

MC Runs167

N comp 6.49

N MS 5.90

N postMS 0.47

N MT

MS

0.26

N MT

postMS

0.23

%MS MT 4.36

%postMS MT50.00

%t comp57.63

%t MS40.87

%t postMS 4.44

%t MT

MS

2.81

%t MT

postMS

0.11

N OTE.—All simulations adopt cluster model

A.MC Runs:number of distinct Monte Carlo sim-

ulations evolved to108yr. N comp :average num-

ber of companions(of any type)per run. N MS &

N postMS :average number of companions of each

type per run.Most postMS companions are captured

while on the MS and do not correspond to differ-

ent stellar companions. N MT

MS & N MT

postMS

:average

number of MS and post-MS companions that undergo mass transfer per run.%MS MT&%postMS MT: number%of MS and post-MS companions that un-dergo mass transfer.%t comp:average%of total simu-lation time spent with a companion of any type.%t MS and%t postMS:average%of time with MS and post-

MS companions.%t MT

MS &%t MT

postMS

:average%of

time spent in MS and post-MS mass transfer phases.

(Kalogera et al.2004;Ivanova&Kalogera2005).In model A clusters with a100M⊙IMBH,the total mass-transfer time fraction is about3%.Of this,about10%is spent under tran-

sient conditions,i.e.,˙M<˙M crit.However,only a very small fraction of the transient mass transfer time,comparable to the transient duty cycle,will be spent in outburst.Observations of Galactic transients imply that this duty cycle is of the or-der of a few percent(McClintock&Remillard2005).Such short duty cycles correspond to very low time fractions of the 108yr cluster age that could be associated with ULX emis-sion:<0.03%.The probability of detecting one of these transient sources in outburst is unfortunately vanishingly low. The other90%of mass transfer time in our simulations is per-sistent and a ULX will result,by de?nition,if the X-ray lu-minosity exceeds1039erg s?1.We?nd that a100M⊙IMBH becomes a persistent ULX for only～2%of total mass trans-fer time in model A clusters;this corresponds to?0.06%of the total age of the young clusters we consider here.We con-clude that either persistent or transient IMBH ULXs would be observable for only a tiny fraction(<0.1%)of the cluster lifetime.

F IG.5.—Distribution of mass transfer time fraction for167standard model simulations with a100M⊙IMBH.

3.4.Averages vs.Medians

It should be emphasized that the mass transfer time frac-tions we present are averaged over all simulations,and that individual simulations are given equal weight,but their re-sults do vary over a wide range.The majority of our simula-tions have no IMBH mass-transfer events at all;it turns out that the median mass-transfer time fraction for each cluster model is always at or very near zero(Fig.5).At the other ex-treme,IMBHs in a few model A simulations acquire very sta-ble mass-transferring companions and spend more than half of their total evolution time in mass transfer phases.It is clear that the averages reported are affected by the presence of these rare outliers,as one can see by comparing the entire model A data set(Table2,t MT MS and t MT postMS)to the smaller set for which triple system data was obtained,which does not include the outliers(Table3,t MT).We would like to note that for the majority of our simulations the predictions for the IMBH bi-naries leading to mass transfer with X-ray luminosities in the ULX regime could be even lower than we report based on the average quantities and results.

3.5.Triple Systems

The signi?cant number of triple systems that form and are arti?cially(although physically self-consistently,i.e.,con-serving energy)disrupted in our simulations warrant a sepa-rate analysis of their possible effect on mass transfer.The NS merger event shown in the example simulation(Figs.3and4), which resulted from an eccentricity boost by a third object,in-dicates that three-body interactions can play an important role even when triples are arti?cially broken.By analyzing the ini-tial orbital characteristics of triples and their coincidence with mass-transfer phases,we develop a sense of whether our treat-ment of triples is,in fact,a reasonable approximation. Almost all of the triples are hard in the outer orbit(see Fig.4),but depending on orbital parameters,the inner binaries may not be in?uenced by the outer companion.We quantify

ULX Sources from Accreting Binaries With IMBH?7

TABLE3

T RIPLE SYSTEMS AND IMBH COMPANIONS

M IMBH=100M⊙

MC Runs109

N trip 3.46

N trip/N bin 0.50

%trip close28.90

%MT aff47.72

%MT close10.24

%t MT 1.60

%t MT

aff

21.07

%t MT

close

4.96

N OTE.—All simulations adopt cluster model A.

MC Runs:number of distinct Monte Carlo simula-

tions for which triples data were obtained(note that

Tables3and5are based on smaller sample sizes than

are other data). N trip :average number of unique

triple systems per run. N trip/N bin :average ratio

of distinct triple systems to distinct binary systems

formed.%trip close:%of triples that are“close”

(?out/a in≤3).%MT aff:%of MT companions

that are potentially affected by one or more tertiary

companions gained during their lifetime.%MT close:

same fraction for close triples.%MT t aff:percent of

total MT time potentially affected by triples.%MT

t close:same fraction for close triples.

the degree of possible companion interaction using the ratio of the outer orbit’s pericenter to the inner orbit’s apocenter (?out/a in),that is,the closest possible approach between the two bodies.The result is shown in Fig.7(upper left window) and in Table3.We assign?out/a in≤3as the critical value of this ratio,below which it is likely that the outer companion will have a nonnegligible effect on the inner binary.These “close”systems comprise less than1/3of all triples. Another diagnostic for triple systems is to calculate the per-centage of mass transfer phases potentially affected.We con-sider“affected”mass transfer as that which follows or coin-cides with triple formation.More importantly,we also cal-culate the percentage of mass transfer phases associated with close triples.While almost half of mass-transferring compan-ions are affected,only～10%are affected by close triples (Table3).If we instead consider the total mass transfer time that is similarly affected,this fraction drops to～5%.

Apart from calculating the fractions of binaries and mass transfer phases that could be affected by our treatment of triples,we also attempt to quantify the possible effects of triple breaking on the mass transfer phases.One way in which an outer companion may affect the inner binary is through the Kozai mechanism(Kozai1962)that can increase the eccen-tricity of the inner binary and lead to mass transfer.We can-not exclude a priori that by breaking the triples we miss some mass transfer events involving the IMBH.We calculate the timescale for the inner binary to reach maximum eccentricity due to the Kozai mechanism using the initial orbital parame-ters of the triples(Innanen et al.1997;Mikkola&Tanikawa 1998;Wen2003):

τkoz?0.16f M i a i 0.5(1?e2o)1.5yr,(7)

where f?0.42ln(1/e i0)/(sin2(i0)?0.4)0.5,e i0and i0are the initial inner eccentricity and inclination,e o is the outer eccen-tricity,M i is the total mass of the inner binary,M o is the outer companion mass,a i and a o are the inner and outer orbital sep-arations,masses are in M⊙,and distances are in AU.Ifτkoz is suf?ciently short,the outer companion left alone could cause the inner binary to undergo mass transfer at pericenter.How-ever,this would be possible only ifτkoz is shorter than the collision timescale,τcoll,appropriate for the triple system:τcoll?3.4×1013×P?4/3d M?2/3tot n?15v?110

× 1+913M tot+ M

8

Blecha et al.

TABLE 4

IMBH BINARY COMPANIONS

M IMBH [M ⊙]50100150200500MC Runs 29167793344 N comp 2.48 6.4910.5212.0322.66 N MS 2.07 5.909.5711.3920.98 N postMS 0.340.470.840.550.98 N MT MS 0.100.260.460.480.91 N MT postMS 0.140.230.380.210.32%MS MT 5.00 4.36 4.76 4.25 4.33%postMS MT 40.0050.0045.4538.8932.56%t comp 24.9857.6370.0476.5189.69%t MS 18.7640.8753.6960.8956.95%t postMS 3.59 4.44 4.05 1.77 2.36%t MT MS 0.21 2.810.76 1.71 2.43%t MT postMS

0.02

0.11

0.16

0.05

0.10

N OTE .—All

simulations adopt cluster model A.Abbre-viations given in Table 2.

F I

G .6.—Increase in companion acquiring rate for a typical 500M ⊙sim-ulation over a 100M ⊙simulation (cf.Fig.1)is shown;the IMB

H has a companion for almost all of its lifetime.Top window:companion mass;middle window:eccentricity;bottom window:orbital separation.

more prevalent at higher masses,this behavior is not as clearly seen for MS or post-MS companions.The evolutionary his-tory of a typical single 500M ⊙simulation is shown in Fig.6to illustrate the large fraction of its lifetime spent with a com-panion.

A clear result in Table 4is that 50M ⊙IMBHs have a very

F I

G .7.—Triple systems formed for 16simulations of 100,150,200,and

500M ⊙IMBHs.The outer orbit’s eccentricity is plotted vs.the ratio of orbital separations for each triple system.Systems for which ?out /a in is ≤3are denoted by open triangles;other triples are closed circles.

low rate of companion capture and mass transfer events.This places a lower limit on the IMBH mass likely to form a lu-minous X-ray binary.For higher IMBH masses,we empha-size as in §3.4that the average mass transfer time fractions are typically dominated by a few simulations with very long mass transfer phases.Consequently,we do not observe any

signi?cant difference in t MT MS or t MT

postMS between 100-500M ⊙IMBHs.

As in 100M ⊙IMBH clusters,X-ray luminosities for other IMBH masses produce transient or persistent ULXs for in-signi?cant fractions of the cluster lifetime,except possibly for the 500M ⊙.Clusters with 500M ⊙IMBHs do spend signi?-cantly more of their mass transfer time as ULX sources;these systems are transient for ?40%of mass transfer time and per-sistent ULXs for ?12%of mass transfer time.Even so,these ULXs would be observable for only ?0.35%of the total clus-ter lifetime.

The rate of triple system formation increases monotonically with IMBH mass (Fig.7,Table 5).Clusters with a 50M ⊙IMBH have very few triple systems,as is expected based on their low rate of IMBH binary formation.500M ⊙IMBHs have by far the highest average rate of triple formation.In fact,more tertiary systems than binary systems form in this case,in a ratio of 1.6:1as compared to 0.5:1for 100M ⊙simulations (Table 5).Our arti?cial treatment of triples limits our ability to analyze reliably clusters with an IMBH 500M ⊙.In the 100-200M ⊙range,we do not ?nd any conclusive pattern in the percent of triples with close orbits,or in the percent of mass transfer affected by triples.

4.2.Cluster Models

To determine the dependence of our results on initial clus-ter conditions,we test two variations of our standard cluster model,as outlined in Table 1.We use our standard IMBH mass of 100M ⊙for all these simulations.

ULX Sources from Accreting Binaries With IMBH?

9

TABLE 5

T RIPLE SYSTEMS AND IMBH COMPANIONS

IMBH Mass [M ⊙]50100150200500MC Runs 20109531639 N trip 0.90 3.467.989.4435.59 N trip /N bin 0.380.500.750.81 1.58%trip close 50.0028.9018.9126.4922.05%MT aff 66.6747.7262.2233.3371.43%MT close 0.0010.2413.33 6.6720.41%t MT 0.32 1.60 1.11 1.00 2.02%t MT aff

3.1321.0741.8225.1667.08%t MT close

0.00

4.96

7.57

0.00

27.14

N OTE .—

All simulations adopt cluster model A.Abbreviations

given in Table 3.

F I

G .8.—Probability density of companion mass at each timestep,shown for cluster models A (167runs),B (102runs),A+(12runs),and B+(19runs).

In model C,when a lower number density is used (n c =104pc ?3),the IMBH has a companion for only ～35%of its evo-lution time,about half the companion time fraction calculated for the higher-density model (Table 6).The IMBH spends only a fraction of a percent of total simulation time with mass-transferring companions (0.53%).We conclude that clusters with our standard core number density,n c =105pc ?3,are much more likely to contain an accreting-IMBH ULX.

IMBH companions in clusters with a ?attened IMF (model B)are slightly more massive,as expected,with most between ～1?10M ⊙(Fig.8).The IMBH has a binary companion for a shorter time fraction (40%)than in model A simulations (Table 6).A decrease is also seen in the mass transfer time fraction between the two models.This could possibly result

TABLE 6

B INARY COMPANIONS OF 100M ⊙IMBH S Model A A+B B+

C MC Runs 167121021937 N comp 6.497.75 4.35 4.32 2.16 N MS 5.90 6.92 3.15 3.05 2.03 N postMS 0.470.500.590.580.13 N MT MS 0.260.500.160.210.05 N MT postMS 0.230.250.210.210.03%MS MT 4.367.23 4.97 6.90 2.67%postMS MT 50.0050.0035.0036.3620.00%t comp 57.6361.4340.3539.8234.71%t MS 40.8748.9018.8823.1932.35%t postMS 4.44 5.56 4.69 2.100.50%t MT MS 2.81 1.430.540.170.52%t MT postMS

0.11

0.05

0.04

0.07

0.01

N OTE .—See Table 1for details of each model.Mod-els A+and B+are identical to A and B,but with 3BBF allowed.Abbreviations given in Table 2.

from having more massive stars in the core,due to the ?atter IMF,that interact with and can disrupt IMBH binaries.How-ever,we caution again that the mass transfer time fraction for our standard model is dominated by a few outliers,so that the difference seen in model B may or may not be signi?cant.The time in which the model B clusters are observable as ULX sources is very short,as in model A clusters:a transient ULX forms for about 3.5%of total mass transfer time,and persistent mass transfer at ULX luminosities occurs for about 7%of mass transfer time.These imply a possible ULX time fraction of <0.05%.Furthermore,the behavior of triple sys-tems in model B is qualitatively and quantitatively similar to that of triple systems in standard model clusters.

Small differences can be seen when 3BBF is allowed in the cluster.3BBF enhances the rate at which massive objects can grow through collisions,since the hard formed binaries can collide with other formed binaries.As a result,companions are a bit more massive in clusters with 3BBF (Fig.8).The difference in the mass distributions is small,but a larger ef-fect is seen in the mass transfer time fraction.The time the IMBH spends with a mass-transferring companion decreases by more than half when 3BBF is included.This phenomenon is probably similar to that which causes the disparity in mass transfer time between model A &B clusters.

Due to the 100%initial binary fraction,we do not need 3BBF to form binaries.Overall,we see that our cluster mod-els are qualitatively similar with and without 3BBF,so we conclude that we can safely omit 3BBF from our simulations.

5.DISCUSSION

We have undertaken the ?rst detailed investigation of IMBH interactions in young (?100Myr),dense cluster cores,in-

10Blecha et al.

corporating binary dynamics with full binary star evolution. Speci?cally,we study the prevalence and characteristics of interacting binaries involving the central IMBH,and we ex-amine and can better understand their role as potential ULX sources.We use numerical cluster simulations that simplify some aspects of detailed dynamical evolution and allow us to focus on individual core stellar interactions while largely ignoring long-term cluster evolution.We analyze our re-sults in terms of the frequency of IMBH binaries formed and their properties,the frequency of associated mass-transferring IMBH binaries,and the associated X-ray luminosities ex-pected.With our cluster model A,we explore an IMBH mass range of50-500M⊙,and we study two variations on this model as well(Table1).Finally,we analyze the nature of triple systems formed as a possible in?uence on our results here and as a guide for designing future simulations.

Our main results are summarized as follows:?IMBHs more massive than100M⊙in model A clusters

spend the majority(>50%)of their evolution time with

a binary companion.

?Mass-transferring IMBH companions,though,are rel-

atively rare(?3%of the IMBH evolution time)com-

pared to the total time the IMBH spends with a com-

panion.

?IMBHs spend more time in mass transfer phases with

MS companions than with post-MS companions.?A much larger number fraction of post-MS companions

than MS companions undergo mass transfer.

?In model A clusters,IMBH binaries are found to be per-

sistent X-ray sources for about90%of the total mass

transfer time,but the X-ray luminosity reaches ULX

values(L X>1039erg s?1)for only～2%of the mass-

transfer time.For the rest of the mass-transfer time

(?10%),IMBH binaries are found to be transients,

which have short duty cycles.Given that the typi-

cal total mass-transfer time is?3%of IMBH evolu-

tion time,these results amount to a vanishingly small

observable-ULX time fraction(<1%)over the lifetime

of the young clusters we consider.

?Higher-mass IMBHs capture more companions

throughout their evolution,making them more likely

in a statistical sense to capture a companion that can

over?ow its Roche lobe.

?Model A clusters with a50M⊙IMBH and cluster

model C have very few mass-transferring IMBH bina-

ries.

?In model B clusters,the IMBH spends less time with

a companion(40%of total time)and less time in mass

transfer(<1%of total time)than in Model A clusters.?For all cluster models and IMBH masses explored here,

the possible ULX time fraction of the cluster lifetime

(100Myr)is found to be typically of～0.1%or even

smaller.

We note that results presented by Hopman,Portegies Zwart,&Alexander(2004)imply a higher incidence of possible ULX activity associated with MS companions to central IMBHs(typically with duration of 107yr).However,these calculations are based on analytical estimates of the possible tidal capture rates for an IMBH in a cluster of single MS stars and the subsequent isolated binary evolution once a MS companion is acquired. These binary evolution calculations are not integrated with any dynamical evolution(analytically or numerically)and, based on our results,we conclude that the assumption of isolated evolution is not realistic.Given the substantial differences in calculation methods,however,it is not possible to make direct and quantitative comparisons between the study by Hopman,Portegies Zwart,&Alexander(2004)and our study.We note that in our simulations we do allow for tidal captures(for details of implementation see Ivanova 2005),but they turn out to be a negligible contributor to the rate at which IMBHs acquire companions(exchanges into binaries is by far the dominant process).

Last,we recognize triple systems as one possibly limiting factor in our simulations.Since triples are arti?cially dis-rupted as they form,we examine the in?uence of triples in terms of their initial orbital parameters and their coincidence with mass transfer phases.We?nd that while a large num-ber of triple systems form per simulation,and while about 1/3?2/3of all100-200M⊙mass-transferring companions do coincide with triple system formation,only a small fraction of mass transfer phases are affected by triples with“close”or-bits.Furthermore,two opposing phenomena related to triple breaking may cause some uncertainties.Mass transfer may be prevented when potential Kozai evolution is arti?cially in-terrupted,or it may be arti?cially induced when release of the outer companion shrinks the inner binary.We?nd that both effects are relevant for only small,roughly equivalent frac-tions of triples,causing a negligible net impact on our mass transfer results.Therefore,we conclude that our treatment of triples in the regime of IMBH masses considered here does not in?uence our conclusions about the possibility of IMBH ULX formation.However,given the higher occurrence of triples in our500M⊙runs,the issues of multiples and cusp formation are most probably more important for more mas-sive IMBHs( 500M⊙)than those considered here.

For these more massive IMBHs,a more physical treatement of multiple systems would in principle allow the remaining uncertainties regarding triples,multiples,and the ultimate for-mation of central cusps to be resolved.However,to examine ULX formation,treatment of the stellar evolution of multi-ple systems cannot be ignored(especially when it involves massive-star companions that lose mass in winds,experience core-collapse and can themselves disrupt orbits and alter dy-namical evolution).Simulations that incorporate all of these effects are simply not possible at present.Instead,for our future explorations of the IMBH-ULX connection in dense clusters,we plan to implement simulations that would repre-sent an improvement over the ones presented here.Such fu-ture calculations could differentiate between triple systems in which the inner binary may be disturbed and those in which the outer companion may be ignored.In the latter case,the third companion would be kept in the system and its dynami-cal evolution alone could be simulated,while full stellar evo-lution and dynamics are calculated for the inner binary.

We thank Marc Freitag for valuable discussions on clus-ter dynamics and cluster properties before and after IMBH formation.This work is supported by a David and Lu-cile Packard Foundation Fellowship in Science and Engineer-

ULX Sources from Accreting Binaries With IMBH?11

ing grant,NASA ATP grant NAG5-13236and LTSA grant NAG5-13056to V.Kalogera,NASA Chandra Theory Award NAS8-03060to N.Ivanova,and NASA ATP grant NAG5-12044to F.Rasio.

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